scholarly journals Stability of observations of partial differential equations under uncertain perturbations

2017 ◽  
Vol 24 (1) ◽  
pp. 45-61 ◽  
Author(s):  
Martin Lazar
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Wave Operator ◽  
System Dynamic ◽  
Schrödinger Equations ◽  
Partial Differential ◽  
Main System ◽  
Different Types ◽  
The Stability ◽  
Observability Estimates

We demonstrate the stability of observability estimates for solutions to wave and Schrödinger equations subjected to additive perturbations. This work generalises recent averaged observability/control results by allowing for systems consisting of operators of different types. We also consider the simultaneous observability problem by which one tries to estimate the energy of each component of a system under consideration. Our analysis relies on microlocal defect tools, in particular on standard H-measures when the main system dynamic is governed by the wave operator, and parabolic H-measures in the case of the Schrödinger operator.

scholarly journals A spectral mapping theorem for semigroups solving PDEs with nonautonomous past

2003 ◽  
Vol 2003 (16) ◽  
pp. 933-951 ◽  
Author(s):  
Genni Fragnelli
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Partial Differential ◽  
The Stability

We prove a spectral mapping theorem for semigroups solving partial differential equations with nonautonomous past. This theorem is then used to give spectral conditions for the stability of the solutions of the equations.

scholarly journals Nonlinear Klein-Gordon and Schrödinger Equations by the Projected Differential Transform Method

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Younghae Do ◽  
Bongsoo Jang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Taylor Series ◽  
Nonlinear Partial Differential Equations ◽  
Differential Transform Method ◽  
Schrödinger Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Differential Transform ◽  
Klein Gordon

The differential transform method (DTM) is based on the Taylor series for all variables, but it differs from the traditional Taylor series in calculating coefficients. Even if the DTM is an effective numerical method for solving many nonlinear partial differential equations, there are also some difficulties due to the complex nonlinearity. To overcome difficulties arising in DTM, we present the new modified version of DTM, namely, the projected differential transform method (PDTM), for solving nonlinear partial differential equations. The proposed method is applied to solve the various nonlinear Klein-Gordon and Schrödinger equations. Numerical approximations performed by the PDTM are presented and compared with the results obtained by other numerical methods. The results reveal that PDTM is a simple and effective numerical algorithm.

Uniform progressing wave solutions of the wave equation

1971 ◽  
Vol 70 (3) ◽  
pp. 455-465
Author(s):  
Erich Zauderer
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Standard Form ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential ◽  
Wave Solutions ◽  
Different Types ◽  
Uniform Expansions

The solution of problems involving the propagation of discontinuities and other singularities for hyperbolic partial differential equations by means of progressing wave expansions is discussed in the book by Courant(l). He also refers to the work of Hadamard, Friedlander, Ludwig and others on this subject. More recently, Ludwig (2), Lewis(3) and others have considered 'uniform' progressing wave expansions for various problems. These expansions are valid in regions where the standard expansions are not suitable and they can be re-expanded in the standard form outside these regions. Examples of such regions are given by envelopes of bicharacteristic curves or, equivalently, caustics and by shadow boundaries such as occur in diffraction problems. In each of these regions, which we term 'transition regions' different types of uniform expansions are required.

scholarly journals The generalized Kudryashov method for the nonlinear fractional partial differential equations with the beta-derivative

2020 ◽  
Vol 66 (6 Nov-Dec) ◽  
pp. 771
Author(s):  
Yusuf Gurefe
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Exact Solutions ◽  
Schrodinger Equations ◽  
Schrödinger Equations ◽  
Fractional Partial Differential Equations ◽  
Partial Differential ◽  
Generalized Kudryashov Method ◽  
Kudryashov Method

In this article, we consider the exact solutions of the Hunter-Saxton and Schrödinger equations defined by Atangana's comformable derivative using the general Kudryashov method. Firstly, Atangana's comformable fractional derivative and its properties are included. Then, by introducing the generalized Kudryashov method, exact solutions of nonlinear fractional partial differential equations (FPDEs), which can be expressed with the comformable derivative of Atangana, are classified. Looking at the results obtained, it is understood that the generalized Kudryashov method can yield important results in obtaining the exact solutions of FPDEs containing beta-derivatives.

scholarly journals A Novel Homotopy Perturbation Algorithm Using Laplace Transform for Conformable Partial Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Sajad Iqbal ◽  
Mohammed K. A. Kaabar ◽  
Francisco Martínez
Keyword(s):  
Partial Differential Equations ◽  
Analytical Solution ◽  
Differential Equations ◽  
Laplace Transform ◽  
Homotopy Perturbation Method ◽  
Approximate Analytical Solution ◽  
Partial Differential ◽  
Homotopy Perturbation ◽  
Approximate Analytical Solutions ◽  
Different Types

In this article, the approximate analytical solutions of four different types of conformable partial differential equations are investigated. First, the conformable Laplace transform homotopy perturbation method is reformulated. Then, the approximate analytical solution of four types of conformable partial differential equations is presented via the proposed technique. To check the accuracy of the proposed technique, the numerical and exact solutions are compared with each other. From this comparison, we conclude that the proposed technique is very efficient and easy to apply to various types of conformable partial differential equations.

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